Modeling transcriptional feedback loops:
The role of Gro/TLE1 in Hes1 oscillations
By Samuel Bernard1 , Branka Čajavec1 , Laurent Pujo-Menjouet2 ,
Michael C. Mackey3∗ and Hanspeter Herzel1
1
Institute for Theoretical Biology, Humboldt University, Invalidenstr. 43, 10115
Berlin, Germany., 2 Vanderbilt University Department of Mathematics , 1326
Stevenson Center, Nashville, TN 37240, USA., 3 Departments of Physiology,
Physics & Mathematics and Centre for Nonlinear Dynamics, McGill University,
3655 Promenade Sir William Osler, Montreal, Qc, Canada, H3G 1Y6.
∗
Corresponding author, M.C. Mackey. Tel: (514) 398 4336, Fax: (514) 398 7452.
[email protected]
The transcriptional repressor Hes1, a basic helix-loop-helix family protein, periodically changes its expression in the presomitic mesoderm. Its periodic pattern
of expression is retained in a number of cultured murine cell lines. In this paper
we introduce an extended mathematical model for Hes1 oscillatory expression that
includes regulation of Hes1 transcription by Drosophila Groucho (Gro) or its vertebrate counterpart, the transducine-like enhancer of split/Groucho-related gene
product 1 (TLE1). Gro/TLE1 is a necessary corepressor required by a number of
DNA-binding transcriptional repressors, including Hes1. Models of direct repression via Hes1 typically display an expression overshoot after transcription initiation which is not seen in the experimental data. However, numerical simulation
and theoretical predictions of our model show that the cofactor Gro/TLE1 reduces
the overshoot and is thus necessary for a rapid and finely tuned response of Hes1
to activation signals. Further, from detailed linear stability numerical bifurcation
analysis and simulations, we conclude that the cooperativity coefficient (h) for Hes1
self-repression should be large (i.e., h ≥ 4). Finally, we introduce the characteristic
turnaround duration, and show that for our model the duration of the repression
loop is between 40 min and 60 min.
Keywords: Hes-1, Gro/TLE1, Hill coefficient, transcriptional repression loop,
turnaround duration
1. Introduction
Recent experiments on cultured mammalian cell lines have shown oscillatory dynamics in the expression of at least three transcriptional factors: Hes1, p53 and
NF-κB (Hirata et al., 2002; Lev Bar-Or et al., 2000; Hoffmann et al., 2002; Lahav
et al., 2004). In this paper we deal with one of them, Hes1. We propose an extended
model for the regulation of Hes1 expression that explains its oscillatory nature, and
look for the common parameter requirements in another published model.
Somites are transient embryonic structures that are formed through segmentation of the presomitic mesoderm (PSM) in a highly regulated process called somitogenesis. They are the origin of the skeletal muscles of the body as well as the axial
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S. Bernard and others
skeleton and the dermis of the back (Christ & Ordahl, 1995; Gossler & Hrabe de
Angelis, 1998). Two molecular systems operate during segmentation. The first one,
called Notch signalling, plays an important role for the synchronization of adjacent
cells through the Notch ligand Delta (forming the Delta/Notch complex) and for
the precise positioning of the segment borders (Jouve et al., 2000). It has been proposed that the Delta/Notch cell-cell signaling pathway sets up cell type boundaries
by modulating cell fate according to the environmental cues (Heitzler & Simpson,
1991). Once stimulated, this complex induces the second molecular system, the
segmentation “clock”. It is an oscillatory mechanism generating periodic waves of
gene expression. The period of these oscillations are short (order of 2 hours). This
gene expression rhythm is called ultradian and can be described as follows.
Once the second molecular system is stimulated, the target genes are expressed.
These genes produce a family of proteins called HES (for Hairy Enhancer of Split)
(Gao et al., 2001). The HES family of proteins are basic helix-loop-helix (bHLH)
type transcriptional repressors. They are primary targets of Notch signaling and
act by negatively regulating transcription of tissue-specific transcription factors
(Ohsako et al., 1994). The mammalian homolog of the ’hairy’ gene in Drosophila,
Hes1, has been found to be essential to neurogenesis, myogenesis, hematopoiesis,
and sex determination (Proush et al., 1994; Ishibashi et al., 1995). Hes1 is a protein
with a very short half-life of ≈30min. It is also known that a single serum treatment
induces an oscillation of Hes1 mRNA and protein concentrations with a 2-hour
period in a variety of cultured cells, such as myoblasts, fibroblasts, neuroblastoma
cells and teratocarcinoma (Hirata et al., 2002).
These oscillations are of particular relevance for somite formation in early development (Schnell & Maini, 2000; Bessho & Kageyama, 2003). The partitioning
of the vertebrate body into a repetitive series of somites requires the spatially and
temporally co-ordinated behaviour of cells in PSM. Hirata et al. (2002) also showed
that Hes1 acts as its own repressor through a negative feedback loop. To account for
the appearance of these oscillations, they proposed the involvement of a third factor
in addition to Hes1 protein and Hes1 mRNA, because three species are necessary
to induce oscillations in a negative feedback system (Griffith, 1968). However, the
introduction of an explicit delay of 15–20 min in a basic two-species model with
negative feedback is also sufficient to explain the oscillations (Jensen et al., 2003;
Monk, 2003). The delay accounts for the time needed for transcription, translation,
and formation of a complex in the nucleus to start repression.
Drosophila Groucho (Gro) and its vertebrate counterparts, the transducine-like
enhancer of split/Groucho-related gene products 1 to 4 (TLEs1 to 4), lack DNA
binding ability but can functionally associate with a number of DNA-binding proteins, including Hes1. Interaction with Gro/TLE1 at the nuclear matrix is necessary
for transcriptional repression by Hes1 (McLarren et al., 2001). Hes1 itself activates
hyperphosporylation of Gro/TLEs bound to it. This correlates with association
with a nuclear matrix, suggesting that chromatin remodeling might be a mechanism for this transcriptional repression (Nuthall et al., 2002). Therefore, we present
here a model that includes Gro/TLE1 in the regulation of Hes1 mediated repression
of transcription.
It is well known that systems of ordinary differential equations (ODEs) with negative feedback loops need to be at least three-dimensional to sustain oscillations.
A requirement for these oscillations is a high cooperativity coefficient, i.e. a large
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The role of Gro/TLE1 in Hes1 oscillations
3
feedback gain is necessary for the onset of oscillations (Tyson & Othmer, 1978).
One such system, the Goodwin model (Goodwin, 1965), has been extensively used
in circadian clock modeling. However, this model requires a cooperativity coefficient
of at least 9 to induce oscillations which is large and unrealistic unless a cascade
mechanism is involved (Ferrell, 1996). On the other hand, the introduction of a time
delay in a feedback loop allows sustained oscillations even in a one-dimensional delay differential equation with a relatively low cooperativity coefficient. Negatively
regulated systems with delays have been used in various biological contexts, from
haematopoietic (blood cell production) diseases to pupil light reflex (Beuter et al.,
2003). In these examples, the relevance of delays is uncontested and their usage has
been of an appreciable utility in explaining nonlinear phenomena. Small gene regulation systems have also been the object of recent mathematical modeling (Santillan
& Mackey, 2004a,b).
With the Hes1 data and system as a guide, this paper considers two different
models of self-repression kinetics, and uncovers conserved features among them.
Section 2 presents an analysis of the two-dimensional system proposed by Jensen
et al. (2003). In §3 we develop an extension of the Jensen et al. model considering
Gro/TLE1 interaction with the Hes1 regulatory pathway. We then discuss the effects of Gro/TLE1 on the properties of oscillatory Hes1 expression after initiation
by a serum shock. Section 4 considers general features of the two models that allow
the estimation of parameter values. We find quantities that are relatively invariant
with respect to different models. We conclude that although interaction of Hes1
with the corepressor Gro/TLE1 is not necessary for oscillatory expression of Hes1,
it does allow a much more realistic activation curve that is consistent with the
experimental data.
2. Two-dimensional Hes1–mRNA repression model
Standard mathematical analysis shows that two-component models with a negative
feedback can not have stable self-sustained oscillations. Two possibilities are then
offered for a simple extension of the model to achieve oscillatory behavior.
In the first, one can invoke the existence of a third dependent variable. This is
the route taken by Hirata et al. (2002) who have considered a three-dimensional
system, in which a third (unknown) species X actively degrades Hes1, while Hes1
also acts as a repressor for X. However, the analysis of this model is not further
considered in this paper. In the second possibility, an explicit delay (Jensen et al.,
2003; Monk, 2003; Lewis, 2003) is considered, which has its origin in the underlying
biology. In either case the Hill coefficient h, which accounts for the cooperativity
of repression activity of agents involved in the inhibition of Hes1 transcription,
is crucial. Although these studies dealt with parameter estimation and stability
analysis, the cooperativity coefficient h was taken as a fixed parameter with a value
between 2 and 4. Takebayashi et al. (1994) showed that Hes1 regulatory region has
four N box sequences and that Hes1 binds to these sequences. Furthermore when
the N box sequences are disrupted, repression activity of Hes1 is severely impaired.
The complexity of transcription regulation does not allow for direct evaluation of
the cooperativity coefficient and thus, the results presented in this section are used
to set plausible ranges on the value of h.
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S. Bernard and others
Table 1. Parameter list
(Parameters used for simulations and analysis. The “m” indicates that the parameter was
estimated from model analysis. References are: 1= (Hirata et al., 2002); 2= (Jensen et al.,
2003; Monk, 2003) and 3= (Bolouri & Davidson, 2003).)
Parameter
α
δ
σ
k
h
u
THopf
τ
Value
0.031
0.029
0.030
—
2∼4
2∼4
90 ∼ 150
10 ∼ 40
Units
min−1
min−1
min−1
au
—
—
min
min
Ref.
(1)
(1)
m
(2)
(2),m
m
(1)
(2),m
f0
1
min−1
(3)
g0
β
1
1
min−1
min−1
—
(3)
Description
Hes1 degradation rate.
mRNA degradation rate.
GroH deactivation rate.
See Appendix A.
Hes1 repression cooperativity. Lower bound.
GroH activation cooperativity. Lower bound.
Period of oscillation of Hes1.
Total delay due to transcription,
translation and transport. Upper bound.
Maximal transcription/translation rate.
See Appendix A.
Maximal phosphorylation rate.
Hes1 translation rate.
A two-compartment model for Hes1–mRNA self-repression with delay can be
written as (Jensen et al., 2003; Monk, 2003; Lewis, 2003),
f0 k h
dmRNA(t)
= h
−δ mRNA(t),
dt
k + Hes1 (t−τ )h
(2.1)
dHes1 (t)
= β mRNA(t) − α Hes1 (t).
(2.2)
dt
Equations (2.1), (2.2) will be referred to as Model A. The first equation describes the
cellular concentration of Hes1 mRNA (mRNA) at time t and the second equation
describes the cellular concentration of Hes1 protein (Hes1 ) at time t. The respective
degradation rates (δ of Hes1 and α of mRNA) and translation rate (β) all have
units of min−1 . The nonlinearity comes from the transcriptional repression activity
of Hes1. The parameters in the feedback loop are the maximum mRNA transcription rate f0 (min−1 ), a DNA dissociation constant k (same units as Hes1 ) and a
Hill coefficient h representing the degree of cooperativity between different factors
intervening in the repression of Hes1 gene by Hes1 protein. The delay τ accounts
for the delay due to Hes1 modification, complex formation and nucleocytoplasmic
molecular transport.
Table 1 gives the values of parameters used for the analysis and simulations.
The details of the computations and analysis are given in Appendix A a.
In this model, a linear stability analysis (cf. Appendix A a) leads to analytical
formulae for local stability connecting the Hill coefficient, Hopf (oscillation) period
and rate constants. Namely, the steady state is locally unstable whenever h is larger
than,
!
!#1/2
"
4π 2
δ
4π 2
α
.
(2.3)
hA ≡
+
+
2
2
αδTHopf
α
αδTHopf
δ
Assuming a period THopf = 120 min, we obtain a lower bound on the cooperativity
A
coefficient hA of hA ' 4.1. The associated critical delay τcrit
can be expressed as,
2
ω
1
A
τcrit
≡ arccos
−
× ω −1 ,
(2.4)
αδh h
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The role of Gro/TLE1 in Hes1 oscillations
(B)
Expression levels
2
0.1
10
-0.3
10
0
0
-0.9
10
100
200
Steady state
Extrema of oscillations
Solutions
0
10
0
-1
10
5
Cooperativity
coefficient h
mRNA
10
15
-2
10
300
400
500
600
10
11
12
Time (min)
(C)
-0.5
10
Period (min)
Hes1
1
0.5
-0.1
10
-0.7
10
mRNA
Hes1
1.5
(A)
5
150
120
90
6
7
8
9
Cooperativity coefficient h
Figure 1. (A) Numerically computed bifurcation diagram of Model A as a function of the
cooperativity coefficient h. The Hopf bifurcation occurs at the intersection of the thick and
dotted lines. The Hill coefficient has an influence on the amplitude of the Hes1 oscillation.
The thick solid lines represent the maxima and minima of the periodic solutions, whereas
the dotted curve represents the steady-state Hes1 and mRNA levels. The thin lines show
two solutions for values of h below and above the bifurcation point. For h = 3, there
is a spiral convergence to a steady state. For h = 10, the solution converges to a stable
limit cycle. (B ) Time evolution of the limit cycle illustrated in panel (A). (C ) Period of
oscillation of the limit cycle as a function of h. In all panels, the parameters are as in table
1, except τ = 11 min.
where ω is the oscillatory angular frequency, ω = 2π/THopf . This leads to a value of
A
the maximal τcrit
, τA ' 19.7 min. (Refer to Appendix A a for a complete analysis.)
When Model A has no explicit delay (τ = 0), its characteristic time scale (CTS),
defined as the inverse of the absolute value of the real part of the eigenvalue λ
(cf. Appendix B), is determined by the decay rates α and δ (Murray, 1993; Strogatz,
1994). Explicitly, CTS = 2 (α + δ)−1 = 33.3 min. Added to τA = 19.7 min, it
gives a characteristic turnaround duration (CTD) of CTD = CTS +τA = 53 min.
The turnaround duration is the time required to have an effective repression after
transcription initiation (see Appendix B for a derivation of these results). As noted
in the Introduction, other proteins with short half-lives are known to have oscillatory
behavior as well, such as p53 and NF-κB.
The value τA constitutes an upper bound on the delay and hA is a lower bound
on the cooperativity coefficient. To have large amplitude oscillations (that is, to
move away from the bifurcation point), it is necessary to have a somewhat smaller
τ and/or larger h. In figure 1, the delay τ was fixed at 11 min, and the bifurcation
diagram is plotted for varying h. The oscillations and the associated period are
shown in the two right hand panels.
3. Gro/TLE1–mediated repression allows tuned response
(a) Gro/TLE1–Hes1 repression model
We now consider the influence of an additional factor known to be involved in the
Hes1 repression loop, namely Gro/TLE1. Protein Gro/TLE1 is activated through
Hes1-induced hyper-phosphorylation. This activation is described by a monotonically increasing Hill function with Hill coefficient u. Moreover, the active form
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S. Bernard and others
phosphorylated
non−phosphorylated
Gro/TLE1
Gro/TLE1
GroH
operator
transcriptional
repression
Hes1
Hes1 gene
Figure 2. Model for Gro/TLE1-mediated Hes1 repression.
of Gro/TLE1 forms a complex with Hes1, denoted GroH , to mediate repression
through a negative feedback loop (cf. figure 2). The variable GroH represents the
repression complex of hyper-phosphorylated Gro/TLE1 with Hes1 protein. The associated equations are written as
and
dHes1 (t)
f0 k h
= h
−αHes1 (t),
dt
k + GroH (t−τ )h
(3.1)
dGroH (t)
g0 Hes1 (t)u
= u
−σGroH (t).
dt
` + Hes1 (t)u
(3.2)
These equations constitute what we call Model B. As with the other two models,
local stability considerations establish relationships between the Hill coefficient,
period, kinetic constants and delay for Model B (cf. Appendix A b). Denoting the
total cooperativity coefficient by c = uh, we obtain an expression similar to equation (2.3):
"
!
!#1/2
4π 2
σ
4π 2
α
c=
+
+
.
(3.3)
2
2
ασTHopf
α
ασTHopf
σ
B
Again, we can calculate the critical delay τcrit
,
2
ω
1
B
τcrit = arccos
−
× ω −1 .
ασc c
(3.4)
The introduction of another nonlinearity does not change the linear stability analysis performed for Model A. It should be noted, however, that this analysis is an
approximation and more detailed numerical analysis shows that the Hill coefficient
threshold hA is slightly higher than predicted (2.5 instead of 2.0 for fixed u = 2.0).
The characteristic turnaround duration has, as for Model A, a value of CTD = 53
min.
Figure 3 (bottom) shows the stability boundary in (h, τ ) space for Models A and
B, while the top part of the figure gives the corresponding Hopf period. The region
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The role of Gro/TLE1 in Hes1 oscillations
7
80
Delay τ (min)
Model A
Model B
60
40
20
0
1
2
3
2
3
4
5
6
7
8
9
4
5
6
7
8
9
300
Period (min)
250
200
150
100
50
1
Cooperativity coefficient h
Figure 3. Comparison between numerical bifurcation analysis of Model A (dashed lines)
and Model B (solid lines). Notice that the small Hill coefficient needed in case of Model
B comes from the fact that the total cooperativity coefficient is h × u. (A), the bifurcation
curves give the delay τ required to induce a Hopf bifurcation for a given h. Crossing the
lines from left to right (or from below) due to parameter variation induces oscillations (see
e.g. figure 1). (B ), associated periods for each bifurcation curve are shown. The grey area
indicates the period interval from 90 to 150 min leading to ranges of the corresponding
Hill coefficients.
of oscillatory expression is above and to the right of the (h, τ ) curves. For Model B,
only the Hill coefficient h is shown. With u = 2, the total cooperativity coefficient is
thus doubled, leading to almost the same result as for Model B. Thus, for a 120 min
period oscillation, the corresponding values of h and τ can be uniquely determined.
For Model B, the Hill coefficient required for 120 min period oscillations is greater
B
than 2.5 and the delay smaller than max(τcrit
) ≡ τB = 20 min. If an interval
between 90 and 150 min for the period is considered, a range for h and τ can be
defined:
– for Model A, 3 < hA < 7, and 15 min < τA < 30 min;
– for Model B, 4 < c < 8, and 10 min < τB < 30 min.
These results show that introducing and new mechanisms does not affect the basic
properties of these models with respect to the Hopf bifurcation. Interestingly, the
model considered by Hirata et al. (2002) consists of three ODEs with two feedback
loops, each with a Hill coefficient of 2 giving a total cooperativity coefficient of 4.
Our comparison of these models reveals general properties of the Hes1 oscillation
that are not yet measured experimentally. In particular, strong cooperativity of
Hes1 repression (h between 3 and 8) and a turnaround duration between 40 min
and 60 min are predicted.
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S. Bernard and others
30
Serum Shock
Model A
Model B
Hes1 expression level
25
20
15
10
5
0
0
120
240
360
480
600
Time (min)
Figure 4. Introduction of a second nonlinearity allows Hes1 expression to rapidly approach
a steady oscillation after initiation by serum shock (solid line), compared with the behavior
of Model A (dashed line).
(b) A second nonlinearity to increase adaptativity
The stability results of the previous sections show that oscillation onset of the
systems depends on model independent quantities such as degradation rates, cooperativity coefficients (or the product thereof), and the characteristic turnaround
duration, including delays due to auxiliary variables. The CTD takes into account
both the time scale introduced by the dynamics of the non-delayed system and the
explicit delay τ . However, the behavior of the solution away from the steady state
can be quite different from model to model.
Experimental data on Hes1 expression level in cultured cells after a serum shock
shows no sign of overshoot in the first cycle of mRNA transcription and Hes1
synthesis (Hirata et al., 2002). On the contrary, the expression levels of both mRNA
and Hes1 rapidly settle down to an oscillatory regime of approximately constant
amplitude. Systems with only one nonlinearity often display a large overshoot before
solutions converge to their attractor. A second nonlinearity is then needed to fine
tune the dynamics of the systems when away from equilibrium. In Appendix C,
using Model A, we compute a lower bound for the value of the Hes1 expression
overshoot with respect to the steady-state value.
From the results of a full numerical simulation of Models A and B, figure 4
compares the initiation of Hes1 synthesis after a serum shock. The serum shock
triggers Hes1 transcription and is modelled by setting Hes1 , mRNA and GroH
initial levels to values close to zero at time 0. In Model A, there is an overshoot due
to the lack of repression mechanisms in the first minutes. In the case of Gro/TLE1mediated repression (Model B), it is assumed that Gro/TLE1 is already active at a
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The role of Gro/TLE1 in Hes1 oscillations
9
low level, and the fast activation rate of Gro/TLE1 supports the rapid convergence
of Hes1 expression to the stable oscillatory pattern.
Even though Model A consists of a nonlinear system with a delayed variable, it
can be solved analytically for a short time given certain initial conditions. It is thus
possible to give a formal bound for the overshoot. The following calculation shows
that the large overshoot displayed in figure 4 is a characteristic feature of Model A.
We assume that before initiation the Hes1 gene is practically silent so the initial
conditions for mRNA and Hes1 are set to zero. Moreover, to simplify the computations, we set α = δ. From table 1 one can see that this is a reasonable assumption.
To obtain a simple expression for the overshoot, we set the transcriptional delay τ
equal to the half-lives of Hes1 mRNA and protein, so τ = ln 2/δ ' 23 min. Calculations given in Appendix C lead to a bound of the overshoot in Hes1 expression of
Hes1 overshoot ≥ (1 − ln 2)βf0 /(2δ 2 ). With a steady state of Hes1 ∗ = β/α the ratio
between the maximal expression level and the steady-state level is
f0
Hes1 overshoot
= (1 − ln 2) .
Hes1 ∗
δ
(3.5)
From table 1, the ratio of maximal transcription to degradation rates, f0 /δ, is large.
For instance, f0 /δ to 30, gives an overshoot about ten times the steady-state value.
It is likely that the ratio f0 /δ is even higher since estimates of 11 initiations per
second have been reported (Bolouri & Davidson, 2003). It is possible to reduce
the overshoot considerably by setting some parameters to unrealistic values. For
example, a translation rate β of one translation every 33 min (0.03 min−1 ) leads to
a smaller overshoot, comparable to Model B simulations. It is likely, however, that β
is much larger, ranging from 1.0 min−1 to 100 min−1 (Ghaemmaghami et al., 2003;
Bolouri & Davidson, 2003). Even though β has no effect on the stability properties
of these models, it is initially of crucial importance after serum shock.
In principle one could replace β in Model A by a nonlinear function without having to take Gro/TLE1–Hes1 interactions into consideration. However, even though
this is biologically plausible, the introduction of this nonlinearity is not motivated
by any experimental data and would be artificial.
Compared to Model A, Model B shows a more realistic activation curve. This is
due to two factors. First, since Gro/TLE1 is a general corepressor, there may exist a
baseline level of phosphorylated protein in the cell before initiation of Hes1 synthesis. This will have an attenuating effect on Hes1 transcription. Second, numerical
simulations show that in Model B a smaller delay τ is needed to generate 120 min
period oscillations due to the introduction of a second nonlinearity. Consequently,
the speed of the response after activation is increased.
4. Discussion
Numerical simulations and analytical results from three models of Hes1 self-regulation
show interesting model-independent features. Encouragingly, the most important
characteristics of the models all lie in the same range. Thus, to be consistent with
the experimental data the cooperativity coefficient associated with Hes1 repression
must be higher than 2, and is likely to be on the order of 4. Further, the explicit
delay included in the model must be of the order of tens of minutes, ranging from
10 min to 50 min depending on the model. The characteristic turnaround duration
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S. Bernard and others
(CTD), however, is less variable. In all model versions, a CTD of about 40 to 60
min is predicted. An explicit delay τ < 20 min seems to be particularly well suited
for two-component regulatory loops. These results are in agreement with previous
studies (Hirata et al., 2002; Jensen et al., 2003; Monk, 2003; Lewis, 2003). Moreover, we have estimated here the value of the cooperativity coefficient, making the
mathematical analysis more complete. Given the complex nature of transcription
regulation, it is likely that this parameter plays an important role in the control of
protein expression.
A linear stability analysis allows one to make estimates of cooperativity coefficients and delays from measured kinetic data, such as Hes1 protein and mRNA
degradation rates and the period of oscillation. The linear analysis also makes it
possible to derive a lower bound for the cooperativity coefficient h and an upper
bound for the delay τ . It is, however, difficult to evaluate the converse: i.e., a lower
bound for τ and an upper bound for h. However, under the assumption that the
system is not too far away from the Hopf bifurcation, the range of periods seen
in different experiments allows us to estimate the range of h and τ . Thus, when
periods are between 90 and 150 min, the total cooperativity coefficient should range
between 3 and 8, with a corresponding delay between 10 and 30 min.
A central quantity introduced in this study is the characteristic turnaround
duration (CTD), defined as the characteristic time between transcription initiation
and repression. The CTD is derived from the linear analysis and should be regarded
as an upper bound. The model-independent estimation of CTD ' 50 min is a
central result based only on partial knowledge of kinetic data. The use of different
models is justified as long as model-independent characteristics of Hes1 dynamics
are conserved. These results are likely to be similar in other negative transcriptional
feedback loops involving, for example, p53, NF-κB or circadian clock genes.
Interaction of Hes1 with Gro/TLE1 is necessary for transcriptional repression
by Hes1 (McLarren et al., 2001). The fact that Hes1 itself activates hyperphosphorylation of Gro/TLEs bound to it suggests a dynamic link between the two
proteins. Introduction of Gro/TLE-mediated repression in a model allows a faster
physiological adaptation after a serum shock and/or Hes1 induction from the Notch
pathway. In feedback loop systems with a single nonlinearity, a large overshoot is
seen after initiation, resulting in an initial strong response followed by smaller oscillations. When the corepressor Gro/TLE1 is introduced, the overshoot is greatly
reduced, and steady oscillatory protein expression levels are observed. Gro/TLE1
is known to be expressed in a variety of cell lines and acts as a general corepressor
(Jimenez et al., 1997). We suggest that the kinetic role of Gro/TLE1 is to fine tune
expression levels after initiation of protein synthesis.
This work was supported by MITACS (Canada) and the Natural Sciences and Engineering
Research Council (NSERC grant OGP-0036920, Canada). We especially thank Drs. M.
Santillán, S. Stifani and R. Kageyama for helpful discussions and comments. This work
was initiated while BČ was visiting the Center for Nonlinear Dynamics, McGill University.
We acknowledge support from the German grant agencies BMBF and DFG.
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The role of Gro/TLE1 in Hes1 oscillations
11
0
Parameter (f0 and k)
10
τ=30
τ=10
τ=20
−1
10
−2
10
2
3
4
5
6
7
8
9
10
Cooperativity coefficient h
Figure 5. Robustness of the bifurcation parameter h with respect to parameters f0 (solid
lines) and k (dashed lines). Different curves are shown for different values of the delay
(τ = 10, 20 and 30 min). Each curve represent the value of either parameter (f0 or k) at
which a Hopf bifurcation occurs when the parameter h is varied. The stability region lies
on the left-hand-side of the curves.
Appendix A. Linear stability analysis
(a) Model A
In this section, we present a detailed linear stability analysis for Model A. The
linear analysis for Model B follows the same scheme, and is outlined below in the
next section.
Equations (2.1, 2.2) are nonlinear differential equations with a delayed negative
feedback. [For a general introduction to delay differential equations and their applications, see Hale & Verduyn-Lunel (1993) and Beuter et al. (2003).] The first step
of the linear stability analysis is the calculation of the steady states. The equations
defining the steady-state are
δ mRNA∗ =
kh
f0 k h
,
+ Hes1 h∗
(A 1)
and
α Hes1 ∗ = β mRNA∗ .
(A 2)
We note that f0 and k have only a minor effect on the bifurcation analysis. Figure 5
shows that even changes of one order of magnitude do not substantially affect the
bifurcation analysis. For example, a change of k from 0.01 to 1 leads to a change of
the bifurcation parameter h from 4.0 to 4.1, at a delay of τ = 20 min. As long as
k takes small values, the parameter k can be freely chosen without perturbing the
Article submitted to Royal Society
12
S. Bernard and others
Theoretical
Numerical
Period (min)
200
150
100
2
3
4
5
6
7
8
10
Theoretical
Numerical
50
Delay τ (min)
9
40
30
20
10
2
3
4
5
6
7
8
9
10
Cooperativity coefficient h
Figure 6. Comparison between theoretical and numerical bifurcation analysis of Model B.
(A) The numerically computed bifurcation curve (dashed-dotted line) shows the value τ at
which a Hopf bifurcation occurs when h is varied. The theoretical bifurcation curve (solid
line) was drawn using Eq. 3.4). Note the good agreement between the numerical and theoretical curves for larger h. (B ) The corresponding period of oscillation (THopf ) is shown.
The numerically computed period (dashed-dotted line) is compared to the theoretically
computed period (solid line) using THopf = 2π/ω with ω defined in Eq. A 18.
bifurcation analysis of Model A. An appropriate scaling is
kα
β
h
=
δ
.
f0 − δ
(A 3)
With this scaling, the steady-state values are mRNA∗ = 1 and Hes1 ∗ = β/α. This
is a reasonable assumption since f0 δ (see table 1), at least for larger h, as
shown in figure 6. Figure 6 shows the comparison between results using the choice
of k discussed above and bifurcation analysis performed with the MATLAB package
DDE-BIFTOOL (Engelborghs et al., 2001). There is an excellent agreement between
the analytical approximation and numerical results.
Linearization of equations (2.1, 2.2) around this steady state yields the following
equations for the deviations from the steady states x = mRNA − 1 and y = Hes1 −
β/α:
dx
= −δx + f∗0 yτ ,
(A 4)
dt
dy
= βx − αy.
(A 5)
dt
Here f is the nonlinear function on the right hand side of equation (2.1) defined by,
f = f0
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kh
kh
.
+ Hes1h
(A 6)
The role of Gro/TLE1 in Hes1 oscillations
Its derivative, evaluated at the steady state f∗0 , is given by
αδh
δ
f∗0 = −
1−
.
β
f0
13
(A 7)
The characteristic (eigenvalue) equation associated with this system of equations is
δ
(λ + δ)(λ + α) + αδh 1 −
exp(−λτ ) = 0.
(A 8)
f0
At the Hopf bifurcation point, a pair of eigenvalues λ has a zero real part, that is,
λ = iω. The value ω gives the frequency of oscillation at the Hopf bifurcation point.
In model A, the frequency is given by,


"
2 #1/2 1/2
1 
δ
 .
(A 9)
ω = √ −(δ 2 + α2 ) + (δ 2 − α2 )2 + 4α2 δ 2 h2 1 −
f0
2
The assumption f0 δ allows us to simplify the expression for hA (equation (2.3)),
which is found by solving equation (A 9) with ω = 2π/THopf . The oscillation onset
(Hopf bifurcation) is given by a sign change of the real part of λ in the characteristic
equation for λ = iω. Then, by solving the characteristic equation of Model A
(equation (A 8)) for τ , we find the relation given in equation (2.4).
(b) Model B
For Model B the steady-state equations are
δ Hes1 ∗ =
f0 k h
,
k h + GroH h∗
(A 10)
g0 Hes1 u∗
.
`u + Hes1 u∗
(A 11)
α GroH ∗ =
Two scalings are possible, for k and `, and we define, k h = α/(f0 − β) and `u =
σ/(g0 −σ). These scalings are valid when f0 α and g0 σ. The steady states will
then be Hes1 ∗ = 1 and GroH ∗ = 1. The linear approximation around the steady
states for the deviations x = Hes1 − 1 and y = GroH − 1 leads to the following
equations:
dx
= −αx + f∗0 yτ ,
(A 12)
dt
and
dy
= g∗0 x − σy,
(A 13)
dt
wherein the derivatives of the nonlinearities at the steady states are
α
,
(A 14)
f∗0 = −αh 1 −
f0
and
g∗0
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σ
= −σu 1 −
g0
.
(A 15)
14
S. Bernard and others
This ultimately leads to the eigenvalue equation
(λ + α)(λ + σ) + Γ exp(−λτ ) = 0,
in which
α
Γ = αhσu 1 −
f0
σ
1−
.
g0
At the Hopf bifurcation, λ = iω gives
i1/2
1 h
ω = √ −(α2 + σ 2 ) + [(α2 − σ 2 )2 + 4Γ2 ]1/2
.
2
(A 16)
(A 17)
(A 18)
Finally, by making the approximation that Γ ' huασ, and solving equation (A 18)
for hu, equation (3.3) and, subsequently, equation (3.4) are obtained.
Appendix B. Characteristic turnaround duration (CTD)
When τ = 0, Model A reduces to a system of ordinary differential equations (ODEs)
with a stable steady state. The linearized equations of Model A become linear
ODEs, and the eigenvalues λ are given by equation (A 8) with τ = 0. That is,
δ
(λ + δ)(λ + α) + αδh 1 −
= 0.
(B 1)
f0
√
Hence, λ = [−(α + δ) ± ∆]/2, with a negative discriminant ∆ < 0 when δ = α.
Thus, the real part of λ is −(α + δ)/2. The turnaround time |λ|−1 + τ gives an
estimation of the time required from mRNA synthesis to repression of transcription,
i.e., a complete turn around.
Figure 7 shows the characteristic turnaround duration Models A and B as a
function of the cooperativity coefficient (h for Model A and c for Model B). The
CTD curves for Model A and B vary are very similar and vary very weakly. Thus,
when h varies from 4 to 10, the CTD only decreases from 53 min to 44 min.
Appendix C. Overshoot and adaptativity
We discuss here the maximal value that solutions of Model A can achieve for initially
vanishing Hes1 mRNA. Numerical simulations show that this maximal value is
reached around t = τ . This will restrict our search to the interval t ∈ (0, τ ] leading
to a lower bound of the maximal value of Hes1 concentration.
Since the non-delayed terms are linear, Model A can be solved directly with
initial conditions given on the interval [−τ, 0]. Setting the initial conditions to zero
yields the following system of linear differential equations
dmRNA(t)
= f0 − δ mRNA(t),
dt
(C 1)
dHes1 (t)
= β mRNA(t) − α Hes1 (t),
dt
(C 2)
with the solutions
mRNA(t) =
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f0
1 − e−δt ,
δ
(C 3)
The role of Gro/TLE1 in Hes1 oscillations
15
CTD A
CTD B
54
CTD (min)
52
50
48
46
44
4
5
6
7
8
9
10
Cooperativity coefficient
Figure 7. Characteristic turnaround durations for Models A and B (solid lines with circles
and x ). For plotting these curves, a period of 120 min was assumed. That is, for a given
h, τ was determined to give a period of 120 min (dashed lines with circles and x ).
δe−αt − αe−δt
βf0
1+
.
(C 4)
αδ
α−δ
These solutions are exact for 0 ≤ t ≤ τ for zero initial conditions. The solution
for Hes1 is a not a convenient expression, but by letting α → δ, and evaluating at
t = τ we have
βf0
Hes1 (τ ) = 2 1 − e−δτ (1 + τ δ) .
(C 5)
δ
The term in parenthesis is a constant of the order of 1. Because estimated values
for τ are of the same order as the Hes1 protein and mRNA half-lives, we take the
delay τ to be equal to the mRNA half-life, τ = ln 2/δ, and we obtain
Hes1 (t) =
Hes1 (τ ) =
βf0
(1 − ln 2).
2δ 2
(C 6)
If we assume that the steady-state Hes1 level is Hes1 ∗ = β/α = β/δ, the ratio
between Hes1 (τ ) and the steady-state Hes1 level can be computed as given in
equation (3.5).
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